Galactic disk flaring is an outward increase of vertical thickness. A constant vertical velocity dispersion in an approximately thin-sheet stellar Newtonian gravitational field gives $h\propto 1/{\Sigma }_{\ast }$. An exponential galactic disk then predicts $h\propto e^{R/R_d}$ for a gas tracer. Inside a thick stellar layer the potential is instead approximately harmonic, and the tracer width is ${\sigma }_z/{\nu }_z$ with vertical epicyclic frequency ${\nu }_z$.

Galactic disk heating increases the random velocities of stars through encounters with molecular clouds, nonaxisymmetric gravitational disturbances, and other perturbations. Older stellar populations usually have larger velocity dispersions and greater vertical thickness, although birth conditions and migration also affect the relation.

A thin galactic disk is a stellar component whose vertical extent is small compared with its radial size. Small vertical velocity dispersion and substantial specific angular momentum are characteristic. Its survival constrains cumulative galactic disk heating, but its stars need not all have the same formation epoch.

The glucagon-like peptide-1 receptor is a G protein-coupled receptor activated by GLP-1, an intestinal peptide signal. In pancreatic beta cells, it can enhance glucose-dependent insulin secretion, prominently through cyclic AMP-linked pathways. It is distinct from the glucagon receptor despite their related ligands and receptor family. Tissue expression and physiological context determine which downstream response dominates. ^[ref]

The glucagon receptor is a G protein-coupled receptor activated by glucagon. In liver cells, signalling prominently increases cyclic AMP and helps favour glucose production, including mobilization of stored carbohydrate. The receptor converts a circulating hormonal signal into intracellular pathway changes. Its response depends on nutritional state and available substrates rather than generating glucose independently of cellular metabolism. ^[ref]

For deterministic initial asset values $A_0$ and a finite-dimensional terminal random vector $A_1$, exactly one possibility holds: a portfolio has nonpositive initial cost and nonnegative payoff with strict gain at at least one date, or there is a strictly positive integrable state-price density pricing $A_1$ as displayed. Apply the strictly positive barycentre cone lemma. Outside the closed support cone, strict separation gives a negative-cost portfolio. On its relative boundary, a supporting functional gives a zero-cost portfolio with positive payoff on a set of positive probability. With a traded positive bank account, normalize its discounted density to obtain an equivalent martingale measure.

For a finite-dimensional discounted gain random vector $X$, absence of arbitrage means $h\cdot X\ge 0$ almost surely only when $h\cdot X=0$ almost surely. Let $N=\{h:h\cdot X=0\text{ almost surely}\}$ and minimize $F(h)=\mathbb{E}{e}^{-|X{|}^2-h\cdot X}$ on the orthogonal complement $N^{\perp }$. The Gaussian damping makes $F$ finite and differentiable without physical integrability assumptions on $X$. Every nonzero direction in $N^{\perp }$ has negative gains on an event of positive probability; compactness of directions proves coercivity. At a minimizer, differentiation gives $\mathbb{E}[X{e}^{-|X{|}^2-h_{\ast }\cdot X}]=0$. Normalizing the positive weight produces an equivalent martingale measure with integrable gains. This is a constructive one-period proof of the fundamental theorem of asset pricing even on an infinite probability space.

Let $Y$ be the discounted gains of finitely many financial assets on any probability space. Tilting the physical probability measure by a normalized multiple of $(1+\Vert Y\Vert {)}^{-1}$ makes $Y$ integrable without changing null events. The positive-weight expectation cone alternative then says either a bounded strictly positive weight makes the weighted gain expectation zero, or a deterministic risky holding gives an arbitrage. Normalizing the weight yields an equivalent martingale measure. Conversely, under such a measure a nonnegative terminal gain with zero initial price has zero expectation, and must vanish almost surely. This proves the one-period fundamental theorem of asset pricing without assuming physical integrability of asset gains.

A factorization tree is a rooted tree whose root is an integer $n>1$. Each nonleaf label is a composite number split into two factors greater than one whose product equals that label; the leaves are prime numbers. Multiplying the leaves recovers $n$. Many factorization trees can have the same root, but the fundamental theorem of arithmetic gives them the same leaf multiset.

The fundamental group of a surface. For a closed orientable genus-$g$ surface the fundamental group of a closed orientable surface has $2g$ generators and one relation, the product of their paired commutators.

For a locally compact Hausdorff space $X$, $C_c(X)$ consists of continuous functions whose support is compact. A positive linear functional on this space, bounded on each fixed compact support, is represented by a Radon measure. For a locally compact group with Haar measure, these functions are dense in $L^p$ for $1\le p<\infty$.

A function is nonconstant when two inputs in its domain of a function have different outputs. For example, $f(x)=x^2$ on the real line is nonconstant, whereas the same rule restricted to $\{-1,1\}$ is a constant function.

A projection is an idempotent self-map $p:X\to X$: applying it twice has the same effect as applying it once, $p\circ p=p$. Equivalently, it fixes every point of its image of a function. The map $p(x,y)=(x,0)$ on the real plane is an example.

A coordinate projection map $X\times Y\to X$ has different source and target, so it is not literally an idempotent self-map. After choosing $y_0\in Y$ and identifying $X$ with $X\times \{y_0\}$, its lifted form $(x,y)\mapsto (x,y_0)$ is a projection. A linear projection additionally respects vector addition and scalar multiplication.

For a scalar-valued function $f$ on a topological space $X$, its support isThe closure matters. On the real line, $f(x)=\max (0,1-|x|)$ is nonzero exactly on $(-1,1)$, but its support is the closed interval $[-1,1]$. A point where $f$ vanishes can still belong to the support if nonzero values occur arbitrarily near it.

For the discrete topology, every subset is closed, so the support is simply the set of inputs with nonzero values. This is the convention used when representing a finitely supported count function. A function has compact support when the displayed closed set is compact.

For a function $f:X\to Y$ and a subset $A\subseteq X$, its restriction $f{|}_A$ keeps the same values but allows only inputs in $A$. Restricting the domain of a function can remove collisions and permit an inverse function, as the nonnegative quadratic function example shows. This changes the allowed inputs; it is different from keeping the inputs and changing the codomain.

Two self-maps $R,S:X\to X$ are commuting functions when their function compositions satisfy $R\circ S=S\circ R$. Order then makes no difference: the self-maps $R(x)=x+1$ and $S(x)=x+2$ on the real line both give $x+3$ when applied successively. By contrast, $R(x)=x+1$ and $S(x)=2x$ give $2x+1$ and $2x+2$, so they do not commute.

Mathematical induction gives $R^n\circ S^m=S^m\circ R^n$ for all nonnegative integers $m,n$, where powers mean repeated function composition. Sharing a fixed point or an invariant set is insufficient to imply commutation. For rational maps of the Riemann sphere with degree of a rational map of the Riemann sphere at least two, commutation does imply equality of their Fatou sets and Julia sets, by the commuting rational maps of degree at least two have the same Julia set theorem.

A real-valued function $f:X\to \mathbb{R}$ is nonnegative when $f(x)\ge 0$ for every input $x\in X$. For example, $f(x)=x^2$ and $g(x)=|x|$ are nonnegative functions on the real line.

Products and nonnegative linear combinations preserve the property: if $f,g\ge 0$ and $\alpha ,\beta \ge 0$, then $f(x)g(x)\ge 0$ and $\alpha f(x)+\beta g(x)\ge 0$ at every point. A partition of unity uses nonnegative functions whose values sum to one, so its weighted combinations lie in the convex hull of the values being combined.

A germ at a point is an equivalence class of functions defined on a neighbourhood of the point: two representatives are equivalent if they agree on some smaller neighbourhood. A germ of a holomorphic function requires its representatives to be holomorphic functions; smooth and continuous germs use their corresponding classes of functions. The germ retains local behavior while forgetting the representative's values farther from the point.

A self-map is a function whose domain of a function and codomain are the same set. It need not be a bijection: a constant function on a two-element set is a self-map that merges its inputs. Self-maps can always be composed with one another, and all self-maps of a specified set form its full transformation monoid.

The codomain of a function is its specified target set. The image of a function is a subset of the codomain, and equals it precisely for a surjective function. Changing a codomain can change whether the same input-output rule is surjective, as the quadratic function examples under function show.